Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Analysis of the Response Matrix for an Extended Target

Abstract: In this paper we study the response matrix obtained from the interelement response of an active array of transducers that can send out signals and record reflected signals. In particular we analyze the eigenvalues and eigenvectors of the response matrix corresponding to the acoustic field reflected by an extended target, the size of which is comparable to the wavelength. We show that the eigenvalues are not well separated for a single extended target in general. However, when both the size of the target and the size of the active array are small compared to the distance from the array to the target, it is shown that the eigenvalues are well separated and that the leading eigenvalues and eigenvectors can be characterized in terms of the location and dimension of the target. Numerical experiments are presented to verify the analysis.

First Colonization of a Spectral Outpost in Random Matrix Theory

Abstract: We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitian matrix model. The method is rigorously based on the Riemann–Hilbert analysis of the corresponding orthogonal polynomials. We provide an analysis with an error term of order N −2γ where 1/γ=2ν+2 is the exponent of non-regularity of the effective potential, thus improving even in the usual case the analysis of the pertinent literature. The behavior of the first finite number of zeroes (eigenvalues) appearing in the new band is analyzed and connected with the location of the zeroes of certain Freud polynomials. In general, all these newborn zeroes approach the point of nonregularity at the rate N −γ , whereas one (a stray zero) lags behind at a slower rate of approach. The kernels for the correlator functions in the scaling coordinate near the emerging band are provided together with the subleading term. In particular, the transition between K and K+1 eigenvalues is analyzed in detail.

The existence of eigenvalues for integral operators

Abstract: Throughout K(x, s) is assumed to be real valued and continuous for A x A (A = [a, b] is a finite closed interval of the real line). For definiteness we fix the domain 9(T) of the integral operator (1) as the Hilbert space L2(A); it should be emphasized at this point that K(x, s) is not necessarily a symmetric kernel. It will be clear from the subsequent analysis that the nature of the spectral set of T is unaltered for any of the alternative specifications g(T)= Lp(A), 1 < p < 00. It is well known that T is completely continuous. Therefore, the eigenvalues A = {)'o2, ...} form a discrete set which may be infinite, finite or empty. Each eigenvalue is of finite algebraic and geometric multiplicity and 0 is the only limit point of {,A} if A is not finite. Finally, the spectrum of the transformation T, apart from point spectrum A, can contain only the origin. Let r(T) denote the spectral radius of T, i.e.,

On the eigenvalues of the matrix

Abstract: Let A A be n×n matrices. We study the eigenvalues of when X runs over the set of n×n nonsingular matrices.